Sharp weak-type inequalities for differentially subordinated martingales

Let $M,N$ be real-valued martingales such that $N$ is differentially subordinate to $M$. The paper contains the proofs of the following weak-type inequalities: (i) If $M\ge 0$ and $0<p\le 1$, then $$\mathrm{\parallel }N{\mathrm{\parallel }}_{p,\mathrm{\infty }}\le 2\mathrm{\parallel }M{\mathrm{\parallel }}_p$$ and the constant is the best possible. (ii) If $M\ge 0$ and $p\ge 2$, then $$\mathrm{\parallel }N{\mathrm{\parallel }}_{p,\mathrm{\infty }}\le \frac{p}{2}(p-1{)}^{-1\mathrm{/}p}\mathrm{\parallel }M{\mathrm{\parallel }}_p$$ and the constant is the best possible. (iii) If $1\le p\le 2$ and $M$ and $N$ are orthogonal, then $$\mathrm{\parallel }N{\mathrm{\parallel }}_{p,\mathrm{\infty }}\le K_p\mathrm{\parallel }M{\mathrm{\parallel }}_p,$$ where $$K_p^p=\frac{1}{\mathrm{\Gamma }(p+1)}\cdot (\frac{\pi }{2}{)}^{p-1}\cdot \frac{1+1\mathrm{/}{3}^2+1\mathrm{/}{5}^2+1\mathrm{/}{7}^2+\mathrm{.}\mathrm{.}\mathrm{.}}{1-1\mathrm{/}{3}^{p+1}+1\mathrm{/}{5}^{p+1}-1\mathrm{/}{7}^{p+1}+\mathrm{.}\mathrm{.}\mathrm{.}}\mathrm{.}$$ The constant is the best possible. We also provide related estimates for harmonic functions on Euclidean domains.